In the below word how many words are there in whic | Permutations and Combinations Questions & Answers

Q:

The Number of times the digit 8 will be written when listing the integers from 1 to 1000 is :

A) 100	B) 200
C) 300	D) 400

Answer & Explanation Answer: C) 300

Explanation:

Since 8 does not occur in 1000, we have to count the number of times 8 occurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form xyz, where $0 \leq x, y, z \leq 9 .$

Let us first count the numbers in which 8 occurs exactly once.

Since 8 can occur atone place in $3 C_{1}$ ways. There are $3 x 9 x 9$ such numbers.

Next, 8 can occur in exactly two places in $3 C_{2} \times 9$ such numbers. Lastly, 8 can occur in all three digits in one number only.

Hence, the number of times 8 occur is

$1 \times (3 \times 9^{2}) + 2 \times (3 \times 9) + 3 \times 1 = 300$

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Q:

If the letters of the word VERMA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word VERMA is :

A) 108	B) 117
C) 810	D) 180

Answer & Explanation Answer: A) 108

Explanation:

The number of words beign with A is 4!

The number of words beign with E is 4!

The number of words beign with M is 4!

The number of words beign with R is 4!

Number of words beign with VA is 3!

Words beign with VE are VEAMR

VEARM

VEMAR

VEMRA

VERAM

VERMA

Therefore, The Rank of the word VERMA = 4 x 4! + 3! + 6 = 96 + 6 + 6 =108

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4 7586

Q:

The number of signals that can be generated by using 5 differently coloured flags, when any number of them may be hoisted at a time is:

A) 235	B) 253
C) 325	D) None of these

Answer & Explanation Answer: C) 325

Explanation:

Required number of signals = $5 P_{1} + 5 P_{2} + 5 P_{3} + 5 P_{4} + 5 P_{5}$

= 5 + 20 + 60 + 120 + 120 = 325

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4 5645

Q:

In how many ways can 100 soldiers be divided into 4 squads of 10, 20, 30, 40 respectively?

A) 1700	B) 18!
C) 190	D) None of these

Answer & Explanation Answer: D) None of these

Explanation:

$100 C_{10} + 90 C_{20} + 70 C_{30} + 40 C_{40} = \frac{100!}{10! x 20! x 30! x 40!}$

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4 9588

Q:

What is the total no of ways of selecting atleast one item from each of the two sets containing 6 different items each?

A) 2856	B) 3969
C) 480	D) None of these

Answer & Explanation Answer: B) 3969

Explanation:

We can select atleast one item from 6 different items = $2^{6} - 1$

Similarly we can select atleast one item from other set of 6 different items in $2^{6} - 1$ ways.

Required number of ways = $(2^{6} - 1) (2^{6} - 1) = {(2^{6} - 1)}^{2}$ = 3969

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1 6947

Q:

If a+b+c =21 what is the total number of positive integral solutions?

A) 109	B) 190
C) 901	D) 910

Answer & Explanation Answer: B) 190

Explanation:

Number of positive integral solutions = $n - 1_{C_{r - 1}}$ = ${}^{20}C_{2}$ = 190

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Q:

If a+b+c = 21, what is the total number of non - negative integral solutions?

A) 123	B) 253
C) 321	D) 231

Answer & Explanation Answer: B) 253

Explanation:

Number of non - negative integral solutions = $n + r - 1 C_{r - 1} = {}^{23}C_{2}$ = 253

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Q:

In how many different ways can 6 different balls be distributed to 4 different boxes, when each box can hold any number of ball?

A) 2048	B) 1296
C) 4096	D) 576

A) 120	B) 180
C) 210	D) 240

In the below word how many words are there in which R and W are at the end positions?

RAINBOW

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In the below word how many words are there in which R and W are at the end positions? RAINBOW

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In the below word how many words are there in which R and W are at the end positions?

RAINBOW